By combining the Extreme Value Theorem and Fermat's Theorem, we
get
Rolle's
Theorem: Let f be a function that is continuous on a
closed interval [a,b] and differentiable on the open
interval (a,b), and suppose that f(a)=f(b)=0. Then there
is a point x=c, somewhere between x=a and x=b, such
that f′(c)=0.
Roughly speaking, the Extreme Value Theorem says that there has
to be a maximum and a minimum. If it's in the interior, Fermat's
Theorem says that it has to be a critical point. Since f is
differentiable on the interior, it has to be a point where
f′(c)=0. The details of this argument, as well as the cases
where the maxima and minima are at the endpoints, are found in the
following video: